Friday, April 1, 2016

I have not been on line for a while. This is really the first day in a while that I have felt like getting on line.

I will say - chemotherapy is different for everyone - I hope my chemotherapy is making my cancer suffer more than it is making me suffer. I'm not complaining (well, yes I am a little) but I'd rather have the chemotherapy than not have it.

Good news though - so far I still have a beautiful head of hair.

I have one three week cycle under my belt. After two more cycles they will repeat my PET scan to see if we are making progress or not.

On the topic of mathematics - I am beginning to learn a bit about medical statistics in general and cancer statistics in more detail. As with most statistics (1) firm answers are never given, and (2) even the partial answers you get are laced with the seeds of misunderstanding. You really have to pay attention to what is said, and also what is not said.

In my last job, before I retired for the second time, I always had a frustrating time trying to show management how to use statistics to find out things that are truly meaningful, and not misleading. It is very hard to explain the significance of a standard deviation to people who barely understand what an average is. And when you can't provide firm answers they tend not to believe what you are providing them.

Anyway, I've got about a week before my next chemotherapy treatment - I will try to get some posts done between now and then.

I will say that I have given a lot of examples of Sequence Numbers, but I'm hoping that maybe some others will jump in and discover some more stuff. I mean, if I can do it, well...

Wednesday, March 16, 2016

I know my posts have be erratic, but I have been dealing with health issues that make it difficult. My hands (and possibly my head) have set up some limitations. I have to work pretty hard to catch my mistakes - and I hope I have been catching most of them.

However, tomorrow I will be starting chemotherapy. I hope it will be successful and will last only a short time, but ... well things don't always go as we hope.

Anyways, I may be off line for a while. But I hope to be back at work soon.

Sunday, February 28, 2016

They are integers that have a special
property.When you calculate the decimal
expansion of the inverse of a Sequence Number you get a recognizable number
sequence (many of these sequences are listed in the Online Encyclopedia of
Integer Sequences ( www.OEIS.org )

What kind of mathematics do I need to know
in order to work with Sequence Numbers?

You need to know how to take the inverse of
an integer.The inverse of 123 is “1
over 123” or “1 divided by 123” (1/123).

You also need to know how to do long
division – really long division that you can’t do on your calculator.But don’t worry – you can do it by hand on
paper OR you can get on the internet and go to ( www.wolframalpha.com )
and use this free “super calculator”.It
will take inputs of about 200 digits, and can provide an output of about 3900
digits.

I would not have been able to do these
calculations without access of the Wolfram Alpha website.And I could not check my answers without the
OEIS website.I would recommend you get
on these websites and play around with them to learn how to use them.

However if you understand how to create the
inverse of an integer, and you understand how to take a fraction and do long
division to get its decimal expansion (how to change a fraction into a
decimal), and you learn how to do
these computations on the internet, then you will have it made.

So, how do I make a Sequence Number that produces a multiplication
sequence?

The secret is to start with a Sequence Number that produces
a counting sequence, but is also divisible by the number that you want to
multiply:

Suppose I want to produce a sequence that
shows the multiples of 3.

998,001 is a Sequence Number that
produces a counting sequence (with terms written in three digit strings), and
it happens to be divisible by 3.

998001/3 = 332667

332,667 is a Sequence Number that
produces a sequence that shows the multiples of 3.

The appendix contains a table that show the
prime factors of all of the Sequence Numbers that produce counting sequences,
with terms written in 1 to 65 digit strings.These Sequence Numbers can be used to produce a multiplication sequence
for any number that is a factor of the counting sequence.You can use a single prime factor or a
combination of prime factors, as long as they are factors of the same Sequence
Number.Let’s try a couple!

99,980,001 is a Sequence
Number produces a counting sequence, with terms written in four digit
strings.Its prime factors are: 3^4 *
11^2* 101^2.

So we can use the is
Sequence number to produce another Sequence Number that produces a
multiplication sequence for 3, 11, 101, 9, 27, 33, 303, etc.

With a little bit of brain power you will
be able to produce multiplication sequences for many prime, and composite,
numbers.

The is one problem that we still need to
address.

All of the Sequence Numbers that produce
counting sequences end with the digit “1”, so none of these numbers are
divisible by 2 or by 5.So we have a
problem – we aren’t able to produce a Sequence Number that shows any the multiplies
any even number or any number that is a multiple of 5.So what do we do?

Suppose I want to
produce a sequence that shows the multiples of 30.The factors of 30 are 2 and 3 and 5.

So first I pick a
Sequence Number that is divisible by 3 (they all are, so I can pick any of
them).

999,998,000,001 is a
Sequence Number that produces a counting sequence, with terms written in six
digit strings.

The magic trick is to
add six “zeros” to the end of the Sequence Number.Adding the zeros makes this number
divisible by 2 and 5

The first 80 non-zero
terms of this sequence are accurately listed above.

(I did do one “trick” to
make this one work.I have to do it
with sequences that multiply by a number that has 2 or 5 as a prime
factor.I will explain it in chapter
two, but for now please understand that it ends up adding an additional zero
term at the beginning, but does not change any of the non-zero terms.For now just trust me and watch in awe at
my wondrous magical abilities with numbers.)

Compare with OEIS
sequence A005843.

Multiples of 3:

333,332,666,667 is a
Sequence Number that produces a decimal expansion the counts by threes, or
shows the threes times table.

The first 80 non-zero
terms in this sequence are shown above.However, this sequence continues much further if you are willing to
follow it.

Compare with OEIS
sequence A008585.

I could choose a
Sequence Number that produces terms written with longer strings, but since my
output is limited to 3900 digits I will actually obtain more terms if we
choose a Sequence Number that produces terms written in four digit stringsl

1/33326667 =

When this expression in
input into Wolfram Alpha, and the maximum number of digits (about 3900
digits), you get the first 650 non-zero terms.

25,640,974,359 is not
the only Sequence Number that produces the 39’s multiplication table.In fact most of the Sequence Numbers listed
in this document have other numbers that will produce similar results, but
using longer or shorter strings to show their results.

25,641,025,640,974,358,974,359
is another Sequence Number that produces the 39’s multiplication table.See the results shown below.

The first 80 non-zero
terms of this sequence are accurately shown above.With additional computing power this list
continues much further, but I have not yet been able to show how far the
sequence continues before the pattern breaks down.

This sequence is not
listed in the OEIS collection.

Multiples of 40:

24,999,950,000,025,000
is a Sequence Number that produces a decimal expansion the counts by 40, or
shows the 40 times table.

2,127,659,574,468,085,106,382,978,723,404,255,319,148,936,169,787,234,042,553,191,489,361,702,127,659,574,468,085,106,383
is a Sequence Number that produces a decimal expansion the counts by 47, or
shows the 47 times table.

Just a thought!If these terms are being written in 46
digit strings then these terms might be accurate up to (or even beyond)
4,700,000,000,000,000,000,000,000,000,000,000,000,000,000,000.

If it is accurate up to
the number shown above that means that we would have the 47 times table with
100,000,000,000,000,000,000,000,000,000,000,000,000,000,000 accurate terms,
each written with 46 digits.

If that is true that we
would have 4,600,000,000,000,000,000,000,000,000,000,000,000,000,000,000
accurate digits after the decimal point!That is more than four quattuordecillion accurate digits.

I have to laugh at my
early training as an engineer struggling to get accuracy of four digits past
the decimal point.Obtaining eight
digit accuracy was impossible because the numbers we were using were not
known with eight digit accuracy.

I even have to laugh at
my fellow mathematicians and computer scientists who endeavor to calculate
pi, or e, or phi to a trillion digits.

But consider the
Sequence Number I used to calculate this sequence.2,127,659,574,468,085,106,382,978,723,404,255,319,148,936,169,787,234,042,553,191,489,361,702,127,659,574,468,085,106,383.It is an integer that is 91 digits long,
but I know that every term after the decimal point is a zero.I know that the next million digits after
the decimal point are all zeros.

This is a simple long
division problem.The number we start
with is kind of big, but the process is still the same.You need a big stack of paper (actually
probably more paper that we have on planet Earth), and you will probably wear
out a bunch of pencils, but if you have the time and determination you can do
this calculation by hand with less that a high school education.

I could increase both
the number of digits per term and the number of accurate terms by using the
Sequence Number: 212,765,957,446,808,510,638,297,872,340,425,531,914,893,
617,021,276,595,744,680,851,063,829,787,234,042,553,191,
489,361,697,872,340,425,531,914,893,617,021,276,595,744,
680,851,063,829,787,234,042,553,191,489,361,702,127,659,
574,468,085,106,383.

17,241,379,310,344,827,586,206,896,548,275,862,068,965,
517,241,379,310,345,000,000,000,000,000,000,000,000,000 is a Sequence Number
that produces a decimal expansion the counts by 58, or shows the 58 times
table.

1,694,915,254,237,288,135,593,220,338,983,050,847,457,
627,118,644,067,796,609,830,508,474,576,271,186,440,
677,966,101,694,915,254,237,288,135,593,220,339 is a Sequence Number that
produces a decimal expansion the counts by 59, or shows the 59 times table.

This sequence number
produces multiples of 123, beginning with 0 * 123, and writing them in 10
digit strings.

Terms are accurate to
the 390th non-zero term (390 * 123 = 47970), which is the limit of my
computation.(This sequence probably
continues to be accurate far after the terms shown above.The reason the calculation stops here is
that is the limit of the output of the software I am using.)

This sequence is not in
the OEIS collection.

Multiples of 137:

72,992,699,270,073 is a
Sequence Number that produces a decimal expansion the counts by multiples of 137,
or shows the 137 times table.

This sequence is a
repeating decimal with a period of 41,958.This means that I would expect to see approximately 6,990 correct terms
in this sequence.I do not have the
computing power at this time to verify this.

This sequence is not
listed in the OEIS collection.

Multiples of 153:

653,594,771,241,829,934,640,522,875,817
is a Sequence Number that produces a decimal expansion the counts by 153, or
shows the 153 times table.

128,462,317,345,377,326,699,999,999,998,715,376,826,546,
226,733 is a Sequence Number that produces a decimal expansion the counts by multiples
of 77.843.839.397, or shows the 77.843.839.397 times table.

About Me

This blog is about a special class of numbers that I call Sequence Numbers. I have been working on them for a few years,and just recently things came together. Phillip is helping me get this material posted.